Computational method for load enhancement factors

ABSTRACT

The computation of a Load Factor, a Life Factor or a Load Enhancement Factor using Modified Joint Weibull Analysis may include retrieving a test data set from at least one database and analyzing the data retrieved for fit with a Weibull distribution model. The test data may be analyzed to determine if at least two coupons have been tested and if applied loads and duration of testing at a component-level were varied. A shape parameter may be calculated for the Weibull distribution model. A scale parameter may be calculated for the Weibull distribution of the data. A stress to life cycle relationship may be calculated to account for scatter in the data through the Weibull distribution data. The Life Factor, the Load Factor or the Load Enhancement Factor may be calculated based on the stress to life cycle relationship to account for scatter.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional of and claims priority toapplication Ser. No. 11/643,960, now U.S. Pat. No. 7,877,229, entitledCOMPUTATIONAL METHOD FOR LOAD ENHANCEMENT FACTORS AND APPARATUS FOREXECUTING SAME filed on Dec. 22, 2006, the entire contents of which isincorporated by reference herein.

FIELD

The present disclosure relates generally to a computational method fordetermining performance of composite structures. More particularly, thepresent disclosure relates to a computational methodology fordetermining a Load Enhancement Factor (LEF) for more accurate and costeffective testing of composite structures and associated software andhardware to execute the methodology.

BACKGROUND

Composite structures are advantageous over metallic structures in anumber of ways, including corrosion resistance, reduced weight, andfatigue performance. However, the modes in which damage initiation andprogression or fatigue occur are complex and extremely difficult tomodel analytically in comparison to traditional metallic structures.

As a result, certifying the durability of composite structures reliesheavily upon experimental fatigue tests. Since fatigue scatter incomposites has been, in general, much higher than in metalliccomponents, experimental fatigue tests must be conducted on manyreplicates to achieve the desired levels of reliability in comparisonwith the same tests when performed on metallic structures. Carrying outsuch a vast array of experimental tests dramatically increases the costand time necessary to complete certification of parts, particularly inthe aerospace industry. In an attempt to account for the effects offatigue scatter and to reduce the time and cost associated with testingcomponents containing composite structures, a number of statisticallybased fatigue testing approaches have been developed. One suchmethodology uses the Load Enhancement Factor (LEF) approach originallydeveloped by the Naval Air Development Center (NADC) in conjunction withNorthrop Corporation in the 1980's in a series of papers oncomputational methods for computing fatigue life and residual strengthin composites and certification methodologies. The tests generallyinclude coupon testing and component testing. A “coupon” is a simplespecimen constructed to evaluate a specific property of a material and a“component” is generally a more complex structure made from thismaterial.

The new LEF approach was developed to overcome the individualdisadvantages of the Load Factor and Life Factor approaches presentedin, for instance, Ratwani, M. M. and Kan H. P., “Development ofAnalytical Techniques for Predicting Compression Fatigue Life andResidual Strength of Composites,” NADC-82104-60, March 1982 and Sanger,K. B., “Certification Testing Methodology for Composite Structures,”Report No. NADC-86132-60, January 1986. The Load Factor and Life Factorapproaches are limited as they require either a high experimental testload which may exceed the static strength of the test material orcomponent or require exceedingly long test periods. The LEF method usesthese factors in concert to achieve both reasonable test durations andtest loads that are well below the static strength of the component.However, the traditional methodology for determining LEF has beenmisused, resulting in the consistent use of an LEF value of 1.15 torepresent all composite materials.

This approach has continued to be used, unmodified, over the past twodecades. However, composite materials and the technology to apply andbuild from them have advanced considerably in the past two decades. Inretrospect, the approach of the traditional method of calculating LEFrelies on many questionable assumptions, is difficult to comprehend, andhas a number of significant limitations. This has resulted in a numberof inconsistencies and inaccuracies in the manner in which LoadEnhancement Factors have been applied to fatigue testing on thecomponent level in the past.

The traditional LEF methodology possesses a number of limitations anddrawbacks, which render it difficult to employ and reduce its accuracyand applicability. The LEF methodology was derived using a “typical”stress-life (S-N) curve as a foundation. The papers outlining thedevelopment of the traditional LEF methodology identify P_(M) as staticstrength. However, the equations used to derive the traditionalmethodology do not make use of static strength, P_(M) in the equations.Therefore, this approach does not account for scatter in staticstrength, an assumption that is never identified by the methodology, butis implicit in the development of the equations. As composites have ahigh degree of scatter in static strength, this is a significantshortcoming of the traditional LEF approach.

Whitehead, et al., entitled “Certification Testing Methodology forComposite Structures,” Vols. I and II, October 1986, (“Whitehead”), apaper setting forth the traditional LEF approach, proposed to computethe LEF through Joint Weibull Analysis, which can be utilized to computethe shape and scale parameters assuming a Weibull distribution bymeasuring scatter within groups, for instance groups of fatigue stresslevels, as set forth in volume I of the paper entitled “CertificationTesting Methodology for Composite Structures”. Coupons must be groupedinto distinct stress levels with many replicates per level. The authorsprovided the following equation listed below and recited as Equation 10on Page 12 of Whitehead Vol. Ito compute the shape (a) and scale({circumflex over (β)}) parameters,

                        (Whitehead  Vol.  I,  Eq.  10)${{\sum\limits_{i = 1}^{M}\left( \frac{\sum\limits_{j = 1}^{n_{i}}{x_{ij}^{\hat{a}}\ln\; x_{ij}}}{\sum\limits_{j = 1}^{n_{i}}x_{ij}^{\hat{a}}} \right)} - \frac{M}{\hat{a}} - \left\lbrack {\sum\limits_{i = 1}^{M}\frac{\sum\limits_{j = 1}^{n_{fi}}{\ln\; x_{ij}}}{n_{fi}}} \right\rbrack} = 0$where:n_(i) (i=1, 2 . . . , M) is the number of data points in the i^(th)group of datan_(f) _(i) (i=1, 2 . . . , M) is the number of failures in the i^(th)group of data.This equation is defined in terms of â so an iterative procedure must beused to arrive at a solution.

Once â is computed, the other Weibull parameters (scale parameters) canbe determined using the following equation listed below and recited asEquation 11 on Page 12 of Whitehead Vol. I,

                        (Whitehead  Vol.  I,  Eq.  11)${\hat{\beta}}_{i} = \left\lbrack {\frac{1}{n_{fi}}{\sum\limits_{j = 1}^{n_{i}}x_{ij}^{\hat{a}}}} \right\rbrack^{\frac{1}{\hat{a}}}$

On the surface, both equations appear to be valid. However, if equation(Whitehead Vol. I, Eq. 10) is dissected and rearranged slightly, theresult is the following,

$\frac{\hat{a}\left\lbrack {{\sum\limits_{i = 1}^{M}\left( \frac{\sum\limits_{j = 1}^{n_{i}}{x_{ij}^{\hat{a}}\ln\; x_{ij}}}{\sum\limits_{j = 1}^{n_{i}}x_{ij}^{\hat{a}}} \right)} - \left\lbrack {\sum\limits_{i = 1}^{M}\frac{\sum\limits_{j = 1}^{n_{fi}}{\ln\; x_{ij}}}{n_{fi}}} \right\rbrack} \right\rbrack}{M} = 1$

In this form, the denominator on the left side of the equation is M, thetotal number of groups—here groups of fatigue stress levels. Therefore,the left side of the equation is merely an average value of M groups.The traditional methodology using Joint Weibull analysis does notidentify the implications of using this simple statistical average inestimating the parameters. However, this method is severely limited. Byusing an average, the equation is only valid for stress levels with thesame number of tested and failed coupons. In addition, this equationimplies that n_(f) _(i) , the number of failures for a given stresslevel, must be equal for all stress levels. This significantly reducesthe applicability of the method.

An S-N curve is typically developed using a number of specimens that aretested over varying stress levels to examine the effect of stress levelon fatigue life. The general pattern of stress-life relationship showsthat as stress increases life decreases, or in other words, higherlevels of stress reduce life expectancy in a component.

However, using the traditional LEF approach and a traditional S-N curve,as previously stated, typically few if any of the data points in thistype of relationship may be utilized to determine an LEF with anaccounting of scatter as there is non-identical stress levels beingtested and, typically, an uneven number of tested and failed couponsexists. To account for scatter in fatigue life, the traditional methodusing Joint Weibull Analysis requires identical specimens be tested atmultiple identical stress levels for the comparison.

Furthermore, although the traditional LEF method clearly states thatenvironmental conditions, specimen geometry, and numerous othervariables affect the Weibull shape parameters, the method still usesthese modal values “for simplicity,” as stated on page 82 of Whitehead,Vol. I. In addition, the reference claims that these modal values are“lower than mean values and, therefore, represent conservative values.”While the traditional LEF method with Joint Weibull Analysis may becorrect in stating that the modal values may be less than the meanvalues for this particular literature review, these values do notnecessarily represent conservative values for all composite materials,environmental conditions, specimen configurations, and other unknownvariables. This is a potentially unsafe assumption.

Using these shape parameters values, the traditional LEF without JointWeibull Analysis method then substitutes them into the followingequation listed below and recited as Equation 17 on Page 46 of WhiteheadVol. II:

                       (Whitehead  Vol.  II,  Eq.  17)$F = \frac{{\mu\Gamma}\left( \frac{\alpha_{R} + 1}{\alpha_{R}} \right)}{\left\lbrack \frac{- {\ln(p)}}{{{\chi_{\gamma}^{2}\left( {2\; n} \right)}/2}\; n} \right\rbrack^{\frac{1}{\alpha_{R}}}}$

This assumes particular modal shape parameters for fatigue life and forresidual strength, a single component-level test for a duration of 1.5lifetimes at the B-Basis level (95% confidence with 90% reliability),the resulting LEF value was computed at approximately 1.15. Despite thewidespread use of this LEF value (1.15), little data exists whichsubstantiates the LEF values that engineers have employed on componenttests. However, it is frequently assumed from the publication of thisinitial NADC value that these values can be repeatedly used and havebeen used on a wide number of composite compositions and structuresbeing tested. In numerous published cases, the computations used toarrive at the LEF, Life Factors, etc. are absent from the documentation.Since many of these cases invoke the use of an LEF of 1.15, it isreasonable to assume that the engineers simply used the shape parametersand LEF values provided in the NADC document without quantifying thescatter of their specific materials under investigation.

However, due to the widespread variability in manufacturing processes,advances in composites, variations and advances in laminate design, andmarked increases in the complexity of theses designs, material types,loading configurations, and the like, the LEF value of 1.15 when used,though assumed initially to be conservative, may in fact beunconservative given the large quantity of unknown variables.

Yet another confusion in the NADC approach is that static strength wasused in the analysis. Residual strength will usually exhibit much morevariability since damage has been induced in the material being tested,unlike static strength, which represents a material in a pristinecondition. Using the static strength to compute a “representative” shapeparameter is contradictory since static strength is never used in thederivation of the traditional LEF equations.

To overcome some of these drawbacks when testing on the coupon-level, ifdata already exists in this form, namely, one replicate per stresslevel, an alternative approach called “Weibull Regression for LEFDetermination” can be utilized. This alternative approach can be used tomodel the S-N relationship, and then to develop LEF under that model.

The shortcomings, assumptions, and omissions of the traditional LEFcomputation methodology are further perpetuated and compounded by theuse of these values in dramatically different geometric configurations,environments, manufacturing methods, and similar variables for compositestructures. For example, some materials, such as aramid (sold under thetrade name KEVLAR), are prone to substantial fatigue scatter whenexposed to environments saturated with moisture.

Furthermore, certain loading conditions are prone to more scatter thanothers are. For example, laminate composites manufacturing using thickbraids or tapes with large tow sizes exhibit more scatter in compressiveloading (due to local buckling effects) than in tension loading. None ofthese factors is accommodated by the traditional method of calculatingLEF. This result is another limitation in LEF analyses derived by thetraditional LEF methodology. The failure to develop LEF analyses thatconsider a number of environmental variables, scatter in staticstrength, and other variables commonly at play in composite structureloading is a significant drawback to current testing practices.

Based upon these weaknesses, the approach that the traditional LEFmethods take in generalizing the Weibull shape parameters and theresulting LEF value of 1.15 is of questionable merit and difficult toaccept given the increasing number of unaccounted variables.Accordingly, the shape parameters and corresponding values should not begeneralized or applied to all composite materials and a morecomprehensive testing methodology is needed.

A new methodology for approaching the computation of the LEFincorporating the characteristics of scatter, residual strength,geometry and environmental variables and scatter in residual strengthand fatigue life is needed to reduce costs, increase safety, andincrease reliability. Due to the inherent cost and extended duration oftesting of composite and metallic structures on the component-level, itis often desired to accelerate testing while still maintaining thedesired level of statistical reliability and confidence. By adjustingboth the load levels and planned duration of a component-level fatiguetest without altering the statistical reliability, both the time andcost of the test may be reduced. Additionally, greater accuracy andhigher levels of safety need to be achieved by a more accurate method ofcalculating the LEF accounting for residual strength and variations inresidual strength through scatter. Accordingly, it is desirable toprovide a method and an apparatus executing the method of improvedcomputation of Load Enhancement Factors that is capable of overcomingthe disadvantages described herein at least to some extent.

SUMMARY

The foregoing needs are met, to a great extent, by the embodimentsdisclosed herein. In one aspect, an apparatus improves the computationof Load Enhancement Factors. According to another aspect, a methodcomputes a value for LEF incorporating the characteristics of scatter,residual strength, geometry, environmental variables and otherheretofore unaccounted variables. Such a method may reduce costs,increase safety, and increase reliability. By adjusting both the loadlevels and planned duration of a component-level fatigue test withoutaltering the statistical reliability, both the time and cost of testingmay be reduced. Additionally, greater accuracy and higher levels ofsafety may be achieved by this more accurate method of calculating theLEF that accounts for residual strength and variations in residualstrength through scatter. Accordingly, a method of improved computationof Load Enhancement Factors and an apparatus for executing the same areprovided.

An embodiment relates to a method of calculating a load enhancementfactor. In this method, a data set may be retrieved from at least onedatabase and the data may then be analyzed for fit with a statisticaldistribution model. Then proceeding to analyze the data retrieved forstatistical conditions. Based on the conditions, calculating a stress tolife cycle relationship accounting for scatter in the test data andcalculating a Load Enhancement Factor based on the stress to life cyclerelationship.

Another embodiment pertains to a method of performing a Modified JointWeibull Analysis, where the method may retrieve test data. The methodmay then analyze the data retrieved for fit with a Weibull distributionmodel for the data and analyzes the test data set retrieved to determineif at least two coupons have been tested and if both the applied loadsand duration of testing at the component-level were varied. The methodmay then compute an one shape parameter for the Weibull distributionmodel for fatigue life data, and one for residual strength data. Themethod may compute at least one scale parameter for the Weibulldistribution of the data and stores the at least one scale parameter.The method may proceed to compute a Life Factor, a Load Factor and aLoad Enhancement Factor based on the shape parameters thereby accountingfor scatter in the test data. The Load Enhancement Factor may bedetermined by setting a confidence level, a reliability level, testduration and the number of component to be tested.

Yet another embodiment relates to a method of performing a Weibullregression analysis. The method may obtain coupon test data and mayapply a Weibull distribution regression function to the coupon test datato estimate stress to fatigue life curve and applies an equation torelate the log of a scale parameter of the Weibull distributionregression function with the log of a stress level of the stress tofatigue life curve. The method may implement an estimation procedure toincorporate the stress to fatigue life relationship in the computationof the Load Enhancement Factor through the intercept and slope of theestimates of the stress to fatigue life curve ({circumflex over (θ)}₀,{circumflex over (θ)}₁). The method may further determine a confidencelevel and using a variance and a co-variance matrix developed from theestimation procedure to compute the Load Enhancement Factor.

The apparatus according to a number of embodiments may include acomputer programmed with software to operate the general purposecomputer in accordance with an embodiment. In some embodiments, theapparatus may include a database with coupon test data and a computerconfigured to apply a Weibull distribution regression function to thecoupon test data to develop an estimated stress to fatigue life curveand apply an equation to relate the log of a scale parameter of theWeibull distribution regression function with the log of a stress levelof the stress to fatigue life curve. The apparatus may implement anestimation procedure to incorporate the stress to fatigue liferelationship in the computation of the Load Enhancement Factor throughthe intercept and slope of the estimates of the stress-to-fatigue lifecurve ({circumflex over (θ)}₀, {circumflex over (θ)}₁); therebyobtaining a confidence level and using a variance and a co-variancematrix developed from the estimation procedure to compute the LoadEnhancement Factor.

There has thus been summarized some of the embodiments in order that thedetailed description thereof herein may be better understood, and inorder that the contribution of any one of the embodiments to the art maybe better appreciated. There are, of course, additional embodiments thatwill be described below and which will form the subject matter of theclaims appended hereto.

In this respect, before explaining at least one embodiment in detail, itis to be understood that the claims herein are not limited in theirapplication to the details of construction and to the arrangements ofthe components set forth in the following description or illustrated inthe drawings. Various embodiments may be practiced and carried out invarious ways. Also, it is to be understood that the phraseology andterminology employed herein, as well as the abstract, are for thepurpose of description and should not be regarded as limiting.

As such, those skilled in the art will appreciate that the conceptionupon which this disclosure is based may readily be utilized as a basisfor the designing of other structures, methods and systems for carryingout the several purposes described herein. It is important, therefore,that the claims be regarded as including such equivalent constructions.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram illustrating steps for processing data tocompute LEF according to an embodiment.

FIG. 2 is a flow diagram illustrating steps for performing a ModifiedJoint Weibull Analysis segment according to an embodiment.

FIG. 3A is a graph illustrating a hypothetical S-N curve with adistribution of data points suitable for use by an embodiment.

FIG. 3B is a table comparing the features of a Load Enhancement Factor(LEF) approach to fatigue testing developed by the Naval Air DevelopmentCenter (NADC)-Northrop to an LEF approach to fatigue testing asdisclosed herein.

FIG. 4 is a flow diagram illustrating steps for performing a WeibullRegression Analysis segment according to another embodiment.

FIG. 5 is a flow diagram illustrating steps for performing a coupontesting segment according to an embodiment.

FIG. 6 is a block diagram of a computer network suitable for use with anembodiment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The following definitions are applicable throughout:

The term “computer” or “computer system” refers to any apparatus that iscapable of accepting a structured input, processing the structured inputaccording to prescribed rules, and producing results of the processingas output. Non-limiting, non-exhaustive examples of a computer include:a computer; a general purpose computer; a supercomputer; a mainframe; asuper mini-computer; a mini-computer; a workstation; a micro-computer; aserver; an interactive television; a hybrid combination of a computerand an interactive television; and application-specific hardware toemulate a computer and/or software. A computer can have a singleprocessor or multiple processors, which can operate in parallel and/ornot in parallel. A computer also refers to two or more computersconnected together via a network for transmitting or receivinginformation between the computers. An example of such a computerincludes a distributed computer system for processing information viacomputers linked by a network.

A “computer-readable medium” refers to any storage device used forstoring data accessible by a computer. Examples of a computer-readablemedium include: a magnetic hard disk; a floppy disk; an optical disk,such as a CD-ROM and a DVD; a magnetic tape; a memory chip or jumpdrive; flash memory; and a carrier wave used to carry computer-readableelectronic data, such as those used in transmitting and receivingpacketized information, e-mail or in accessing a network.

The term “Software” refers to prescribed rules to operate a computer.Non limiting, non-exhaustive examples of software include: software;code segments; instructions; computer programs; and programmed logic.

A “computer system” refers to a system having a computer, where thecomputer comprises a computer-readable medium embodying software tooperate the computer.

A “network” refers to a number of computers and associated devices thatare connected by communication facilities. A network may includepermanent connections such as cables and/or temporary connections suchas those made through telephone or other communication links. Examplesof a network include: an internet, such as the Internet; an intranet; alocal area network (LAN); a wide area network (WAN); and a combinationof networks, such as an internet and an intranet.

An “information storage device” refers to an article of manufacture usedto store information. An information storage device has different forms,for example, paper form and electronic form. In paper form, theinformation storage device includes paper printed with the information.In electronic form, the information storage device includes acomputer-readable medium storing the information as software, forexample, as data.

The term “A-basis” refers to the 95% confidence limit for 99%reliability or a value above which at least 99% of the population of thevalues are expected to fall, with a confidence of 95%.

The term “B-basis” refers to a 95% confidence limit for 90% reliabilityor a value above which at least 90% of the population of values isexpected to fall, with a confidence of 95%.

The following abbreviations are used herein:

B(S) represents a B-basis value in fatigue cycle at stress S.

B(X/S) represents the B-basis value in residual strength at X run-outcycles and stress S.

E(W/X,S) represents the expected residual strength at X run-out cyclesand stress S.

E(X/S) represents the expected fatigue failure cycle at stress S.

f(.) represents a probability density function.

F(.) represents a cumulative probability function (also called adistribution function).

i represents a Group number corresponding to a particular stress levelfor coupon-level testing (used in Joint Weibull analysis) e.g. i=1, 2,3, . . . .

j a Data point (element) within i^(th) group.

LEE represents Load Enhancement Factor.

M represents Total number of groups (stress levels) for coupon-leveltesting.

n₀ represents Number of planned component-level tests (sample size usedin future test planning)

n_(f) _(i) represents Number of coupons failed during fatigue in thegroup (stress level).

n_(i) represents the number of coupons in the group (stress level).

n_(L) represents the total number of coupons used in the Weibull modelfor fatigue life cycle (total number of coupons tested in fatigue).

n_(r) represents the total number of coupons used in the Weibull modelfor residual strength.

n_(r) _(i) represents the number of coupons used in the Weibull modelfor residual strength in the i^(th) group (stress level); number ofcoupons surviving the fatigue test to run-out.

n_(s) represents the number of coupons tested for static strength.

N represents the design life (in number of lifetimes) at the desiredreliability.

N₀ represents the test duration (in number of lifetimes) forcomponent-level testing.

N_(i) represents the design life at desired reliability for i^(th)stress level.

N_(F) represents the Life Factor derived from coupon-level fatiguetesting.

N_(M) _(i) =E(X/S) represents the mean life for each stress level forcoupon-level testing; keep it but define it as equal to E(X/S).

P_(T) _(i) represents the mean residual strength per stress level.

R(.) represents the reliability function; desired level of reliability.

S represents the stress level applied in fatigue test or staticstrength.

S₁ represents the fatigue stress level applied to component to achievedesired reliability with test duration of 1 lifetime.

S₂ represents the fatigue stress level applied to component to achievedesired reliability with test duration of equal to the Life Factor.

S_(A) represents the stress level computed from the A-basis value of thefatigue cycle distribution.

S_(B) represents the stress level computed from the B-basis value of thefatigue cycle distribution.

S_(E) represents the stress level computed from an expected value of thefatigue cycle distribution.

S_(F) represents the load factor.

S_(j) represents the static strength of j data point for coupon-leveltesting.

S_(M) represents the mean static strength.

S_(r) represents the static strength allowable.

W represents the residual strength at X run-out cycle and stress S.

W_(r) _(i) represents the residual strength allowable for i^(th) stresslevel.

X represents the number of fatigue cycles to failure or to “run-out” fora coupon.

X₁ represents the number of cycles on the coupon level that defines onelifetime.

Y=ln(X) represents the natural logarithm of fatigue cycle.

α represents the scale parameter of a two-parameter Weibulldistribution.

α_(Li) represents the scale parameter of the Weibull model for fatiguelife measured from coupon-level testing for i^(th) group (stress level).

α_(r) _(i) represents the scale parameter of the Weibull model forresidual strength measured from coupon-level testing for i^(th) group(stress level).

α_(s) represents the scale parameter of the Weibull model for staticstrength measured from coupon-level testing.

β represents the shape parameter of a two-parameter Weibulldistribution, also known as the Weibull slope.

β_(L) represents the shape parameter of the Weibull model for fatiguelife measured from coupon-level testing.

β_(r) represents the shape parameter of the Weibull model for residualstrength measured from coupon-level testing.

β_(s) represents the shape parameter of the Weibull model for staticstrength measured from coupon-level testing.

β_(SN) represents the shape parameter for Weibull regression model.

δ represents the dimensionless parameter equal to 1 if X is a fatiguefailure, equal to 0 if X is a “run-out”.

γ represents the confidence level.

Γ represents the gamma function.

K(N₀) represents the scaling coefficient in the Load Enhancement Factorto ensure that

LEF=1 when the test duration is N_(F).

θ₀ represents the intercept parameter of the Weibull regression model.

θ₁ represents the slope parameter of the Weibull regression model.

χ_(γ,ν) ² represents the Chi-square random variable with γ confidencelevel and ν degrees of freedom.

An embodiment includes a method of computing load enhancement factorsand an apparatus for performing the same. The application of loadenhancement factor involves raising the stress level in test andreducing test duration in such a way that a desired reliability isdemonstrated when no failure is observed. Testing under a specificstress level and duration without failure can demonstrate certainreliability level. By raising the stress level and shortening theduration, time and testing costs may be saved without lowering theintended reliability level. Other applications of the load enhancementfactor involves back calculating the reliability level for a given valueof Load Enhancement Factor (LEF) and duration, or back calculating theduration for a given value of LEF and reliability level. These two backcalculations are used to assess and compare existing or new testprograms.

FIG. 1 is a flow diagram illustrating steps for executing a method ofdetermining a LEF according to an embodiment. The method steps includestarting the analysis by verifying completion of testing at method step120 and retrieval of data from a database 100 at method step 150. Theretrieved data or information from database 100 may include, but is notlimited to, coupon level testing results from tests run on subjectcomposite coupons, time stamp information, test item identification. Thecoupon testing may be conducted in any manner that is appropriate todeveloping the necessary data set. Variables that may be incorporated inthe grouping of the data can include, but are certainly not limited to,stress levels, life cycles, environmental conditions, coupon size,coupon shape, coupon manufacturing process, and similar variables.

An analysis of the data is performed, determining whether the couponlevel testing was conducted with a single stress value or whethermultiple stress levels were tested at method step 200. If a singlestress value was tested, the flow chart proceeds in the negative to step420 and progresses therefrom. If multiple stress values were tested, theflow chart proceeds in the affirmative to step 430 and proceedstherefrom. Although the example shown in FIG. 1. utilizes the Weibullstatistical distribution model, similar analytic steps may be used toanalyze data fitting different distribution models, including but notlimited to the normal statistical distribution and similar statisticaldistributions. Adjustment of the equations is hereby contemplated toaccommodate such distribution models to accommodate scatter and theother variables contemplated above in the coupon testing or other datastored on the database 100 and such further developments are within thespirit of the various embodiments.

In the case of a single stress value being tested, the Joint Weibullcode segment 1000, see for instance FIG. 3A, may be applied alone or incombination with the Weibull regression code segment 2000, see forinstance FIG. 4, depending on the analysis of the data, to determine aLoad Enhancement Factor (LEF). In method step 420 and 430, a Weibulldistribution is assumed and determination of sufficient failure data iscarried out. The next determination is made in method step 440 whethersufficient runout data points exist in the data set from database 100.If the runout data is insufficient, then the method proceeds in thenegative and ends.

If sufficient runout data exists, then the method proceeds in theaffirmative to method step 460. At method step 460, a determination ismade whether the data on static strength in the coupon testing isincluded. If the static data is not used, the method continues in thenegative to the Modified Joint Weibull Analysis code segment 1000 andcontinues computations through this methodology only. If static strengthdata is used here only to aid the Weibull regression approach because ofonly single stress data. The method continues to the Weibull RegressionAnalysis code segment 2000. It is understood that the computer canattempt to compute an LEF value for both cases or elect to compute usinga single method. The results are displayed in method step 500. The usermay then elect to use the LEF that, to the best engineering judgment ofthe user, that best fits the goals of the investigation.

The determination of which LEF value to utilize is, in this case, leftto the best engineering judgment of the user. This engineering judgmentmay be based on the type of data available and additional factors thatmay include, but certainly are not limited to, the type of compositematerial, the operational environment of the material, the configurationof the composite material, the manufacturing process of the compositematerial, the testing being conducted at the component level on thematerial, and similar variables. It is also understood that a furtherlogic may be added to exclude one or the other of the computations basedon further qualitative inputs from the user that would typify suchfactors.

If multiple stress values were tested, the flow chart proceeds at step200 in the positive to step 430. In method step 430, the Weibulldistribution is assumed. A determination is made in method step 430 if asingle replicate was tested or multiple replicates were tested in thetest data retrieved in step 150 from database 100. If a single replicatewas tested, the method proceeds in the affirmative at step 450 todetermine if there are sufficient run-out in each stress level. Themethod proceeds to the Weibull Regression code segment 2000 if there aresingle replicate per stress level and multiple stress level. The methodproceeds to computer LEF after fitting the Weibull regression model. TheLEF computation is then output to the user in method step 500.

If multiple replicates were tested in the test data retrieved in step150 from database 100, then the method proceeds to steps 440 and 450when there are enough failure data and run-outs at multiple stresslevel. Both the Modified Joint Weibull Analysis and Weibull regressionmodel would be implemented, and LEF from both approaches would bereported in step 500. The user may then elect to use the LEF that, tothe best engineering judgment of the user, that best fits the goals ofthe investigation.

The determination of which LEF value to utilize is, in this case, leftto the best engineering judgment of the user. This engineering judgmentmay be based on the type of data available and additional factors thatmay include, but certainly are not limited to, the type of compositematerial, the operational environment of the material, the configurationof the composite material, the manufacturing process of the compositematerial, the testing being conducted at the component level on thematerial, and similar variables. It is also understood that a furtherlogic may be added to exclude one or the other of the computations basedon further qualitative inputs from the user that would typify suchfactors.

As noted, the primary drawbacks of using the Whitehead, Vol. Itraditional LEF computation approach described earlier included thelimitation that requires an equal number of coupon replicates be testedat every stress level and the further requirement that an equal numberof replicates survive to run-out at every stress level. The ModifiedJoint Weibull Analysis and Weibull Regression methods presented hereinovercomes, at least, these limitation. It allows for the analysis of thedata derived from coupon testing to determine what scenario best suitsthe data and applying the code segment for Modified Joint WeibullAnalysis, the code segment for Weibull Regression, or both codesegments.

FIG. 2 shows a flow chart for the method of calculating LEF in aModified Joint Weibull Analysis. The methodology uses a Modified JointWeibull Analysis. The Modified Joint Weibull Analysis allows for the useof data derived from an unequal number of coupon replicates that aretested within a variety of stress levels in method step 1020, as shownin FIG. 2. The data from the coupon testing code segment 1010 utilizestesting of coupons to define Weibull parameters. The Modified JointWeibull analysis presented here is versatile and more robust than thetraditional method described in Whitehead et al. reports, providing thefreedom to run as few or as many test replicates within each stresslevel as deemed necessary. The foundation to formulate the approachremains the same as that used by the traditional LEF method, howeversignificant improvements are made to facilitate the inclusion of scatterand residual strength in calculating the LEF, thus overcoming theshortfalls of the traditional method.

The method may utilize various data points and values develop and storedfrom the coupon-level testing segment 1010 for each configuration typeunder investigation. This data and values can include, but are notlimited to: Weibull shape parameters for both fatigue life and residualstrength; Weibull scale parameters for both fatigue life and residualstrength; and Stress-Life relationships in the form of S-N curves.

To compute the Weibull shape parameter for fatigue life, ( ) theequation for the Modified Joint Weibull analysis uses EQ-1 in methodstep 1010,

$\begin{matrix}{{{\sum\limits_{i = 1}^{M}\left( {n_{fi}\frac{\sum\limits_{j = 1}^{n_{i}}{X_{ij}^{\beta_{L}}{\ln\left( X_{ij} \right)}}}{\sum\limits_{j = 1}^{n_{i}}X_{ij}^{\beta_{L}}}} \right)} - {\sum\limits_{i = 1}^{M}\frac{n_{fi}}{\beta_{L}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{fi}}{\ln\left( X_{ij} \right)}}}} = 0} & \left( {{EQ}\text{-}1} \right)\end{matrix}$Note that n_(f) _(i) does not equal n_(i) since some of n_(i) couponsmay survive to “run-out.”

To compute the Weibull shape parameter for residual strength, (β_(r)),the method uses equation EQ-2, similar to equation EQ-1, in method step1015 and n_(r) _(i) is used for the Modified Joint Weibull analysis andbecomes,

$\begin{matrix}{{{\sum\limits_{i = 1}^{M}\left( {n_{r_{i}}\frac{\sum\limits_{j = 1}^{n_{i}}{W_{ij}^{\beta_{r}}{\ln\left( W_{ij} \right)}}}{\sum\limits_{j = 1}^{n_{i}}W_{ij}^{\beta_{r}}}} \right)} - {\sum\limits_{i = 1}^{M}\frac{n_{r_{i}}}{\beta_{r}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\ln\left( W_{ij} \right)}}}} = 0} & \left( {{EQ}\text{-}2} \right)\end{matrix}$Noting that n_(r) _(i) represents the number of coupons undergoingresidual strength tests. These are the “run-out” coupons from the i^(th)group (stress level). The Weibull shape parameter for static strength,(β_(s)), is computed in equation EQ-3 an analogous manner at step 1017,namely,

$\begin{matrix}{{{n_{s}\frac{\sum\limits_{j = 1}^{n_{s}}{S_{j}^{\beta_{s}}{\ln\left( S_{j} \right)}}}{\sum\limits_{j = 1}^{n_{s}}S_{j}^{\beta_{s}}}} - \frac{n_{s}}{\beta_{s}} - {\sum\limits_{j = 1}^{n_{s}}{\ln\left( S_{j} \right)}}} = 0} & \left( {{EQ}\text{-}3} \right)\end{matrix}$These equations must be solved using an iterative approach. Generally,the higher the shape parameter, the less variability a materialexhibits.

Note that these equations for the Modified Joint Weibull analysis aredifferent from those presented by the traditional LEF method.Specifically, these equations, in conjunction with the computations thatfollow, overcome the requirements that an equal number of couponreplicates be tested at every stress level and that an equal number ofreplicates survive to run-out at every stress level. They do not simplyaccount for a simple statistical average, as noted above regarding thetraditional method of calculating LEF. Instead, the Modified JointWeibull Analysis equations accommodate scatter in the data throughoutthe tested groups, regardless of the presence of equal numbers of testedcoupon replicates.

The relevant equations allow an unequal number of coupon replicates tobe tested within each stress level, allowing the computation to quantifyand utilize scatter within the residual strength testing and data. Theaccommodation of scatter is evident from the test data developed inmethod step 1010. A sample of an S-N curve is shown in FIG. 3A. The S-Ncurve is shown using hypothetical values as an example of the resultscontemplated by the iterative approach employed on the equations EQ-1through EQ-3. The analysis presented by the traditional method, ifanalyzing the hypothetical data points of FIG. 3A, would allow forconsideration of a maximum of 6 data points to be used (at a stresslevel of 105 thousand pounds per square inch (“ksi”)). By comparison,the modified equations EQ-1 through EQ-3 would allow 20 data pointscomprising data from a variety of stress levels. Note that stress levelscontaining only one coupon replicate data point cannot be utilized toquantify scatter.

After both shape parameters have been computed, the scale parameters canthen be determined in method step 1045. Since a unique scale parameterexists for each stress level, there will be, at most, M scale parametersfor both fatigue life and residual strength. By definition, each of thescale parameters represents the value at which 63.2% of all data pointsshould fall below. For example, suppose the life scale parameter for aparticular stress level is α_(L)=567,890 cycles. This implies thatapproximately 63.2% of the coupons tested at this stress level will failprior to reaching 567,890 cycles. Similarly, suppose the scale parameterfor residual strength for a particular stress level is α_(r)=58.3 ksi.This equates to about 63.2% of the coupons that are tested for residualstrength will have a residual strength less than 58.3 ksi.

The life scale parameter for each stress level (i^(th) group) iscomputed using equation EQ-4 in step 1045,

$\begin{matrix}{\alpha_{L_{i}} = \left\lbrack {\frac{1}{n_{f_{i}}}{\sum\limits_{j = 1}^{n_{i}}X_{ij}^{\beta_{L}}}} \right\rbrack^{\frac{1}{\beta_{L}}}} & \left( {{EQ}\text{-}4} \right)\end{matrix}$While the scale parameter for residual strength for each stress level(i^(th) group) is computed using equation EQ-5 in step 1047,

$\begin{matrix}{\alpha_{r_{i}} = \left\lbrack {\frac{1}{n_{r_{i}}}{\sum\limits_{j = 1}^{n_{i}}W_{ij}^{\beta_{r}}}} \right\rbrack^{\frac{1}{\beta_{r}}}} & \left( {{EQ}\text{-}5} \right)\end{matrix}$Similarly, the scale parameter for static strength can be computed inequation EQ-6 in step 1049,

$\begin{matrix}{\alpha_{s} = \left\lbrack {\frac{1}{n_{s}}{\sum\limits_{j = 1}^{n_{s}}S_{j}^{\beta_{s}}}} \right\rbrack^{\frac{1}{\beta_{s}}}} & \left( {{EQ}\text{-}6} \right)\end{matrix}$

Once the shape and scale parameters have been established in equationsEQ-1-EQ-6, values for the Chi-squared distribution and Gamma functionsmay be obtained. The value of the Chi-squared distribution establishingγ level of confidence is, χ_(γ,2n) _(L) ², where 2n_(L) are the degreesof freedom corresponding to coupon-level testing.

The equation used to compute the value of the Gamma function for thelife shape parameter is provided in equation EQ-land computed in step1050,

$\begin{matrix}{\Gamma\left\lbrack \left( \frac{\beta_{L} + 1}{\beta_{L}} \right) \right\rbrack} & \left( {{EQ}\text{-}7} \right)\end{matrix}$Values of the Gamma function are computed for varying life shapeparameters can be computed and stored in a database in, for example, aseries of tables. Method step 1055 includes a lookup step, wherein thevalue of the gamma function is determined from those stored in a table.Similarly, the equation used to compute the value of the Gamma functionfor the residual strength shape parameter is provided as EQ-8,

$\begin{matrix}{\Gamma\left\lbrack \left( \frac{\beta_{r} + 1}{\beta_{r}} \right) \right\rbrack} & \left( {{EQ}\text{-}8} \right)\end{matrix}$Values of the Gamma function are computed for varying residual strengthshape parameters are similarly stored and the method step 1055 alsoprovides for a lookup of the gamma value.

Likewise, the mean static strength is computed, the mean residualstrength at the stress level, and mean fatigue life at the i^(th) stresslevel are computed, the mean static strength being computed in step 1060using EQ-9,

$\begin{matrix}{S_{M} = {\alpha_{s}{\Gamma\left\lbrack \left( \frac{\beta_{s} + 1}{\beta_{s}} \right) \right\rbrack}}} & \left( {{EQ}\text{-}9} \right)\end{matrix}$the mean residual strength at the i^(th) stress level being computed atstep 1065 using EQ-10,

$\begin{matrix}{P_{T_{i}} = {\alpha_{r_{i}}{\Gamma\left\lbrack \left( \frac{\beta_{r} + 1}{\beta_{r}} \right) \right\rbrack}}} & \left( {{EQ}\text{-}10} \right)\end{matrix}$the mean fatigue life at the i^(th) stress level being computed at step1069 using equation EQ-11,

$\begin{matrix}{N_{M_{i}} = {\alpha_{L_{i}}{\Gamma\left\lbrack \left( \frac{\beta_{L} + 1}{\beta_{L}} \right) \right\rbrack}}} & \left( {{EQ}\text{-}11} \right)\end{matrix}$

Similarly, the design life at a desired reliability (R) and the residualstrength allowable at a desired reliability (R) for the i^(th) stresslevel are computed in method step 1070 using equations EQ-12 and EQ-13,respectively, where EQ-12 is used in step 1070 to compute the designlife at the desired reliability,

$\begin{matrix}{N_{i} = {\alpha_{L_{i}}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2\; n_{L}}}^{2}}{2\; n_{L}}} \right\rbrack}^{\frac{1}{\beta_{L}}}} & \left( {{EQ}\text{-}12} \right)\end{matrix}$Note that the sample size in the denominator refers to the total numberof coupons tested in fatigue. This accounts for both failures andrun-outs, since these lives are still considered valid data points.

The residual strength allowable at desired reliability (R) for thei^(th) stress level is computed in step 1070 using EQ-13,

$\begin{matrix}{W_{r_{i}} = {\alpha_{r_{i}}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2\; n_{r}}}^{2}}{2\; n_{r}}} \right\rbrack}^{\frac{1}{\beta_{r}}}} & \left( {{EQ}\text{-}13} \right)\end{matrix}$The static strength allowable at desired reliability (R) is computed inclaim 1060 using equation EQ-14,

$\begin{matrix}{S_{r} = {\alpha_{s}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2\; n_{s}}}^{2}}{2\; n_{s}}} \right\rbrack}^{\frac{1}{\beta_{s}}}} & \left( {{EQ}\text{-}14} \right)\end{matrix}$Note that the sample size in the denominator refers to the total numberof coupons reaching the run-out condition. Since residual allowable onlypertains to residual strength, only the coupons reaching run-out arecounted in the sample size.

The method proceeds to utilize the test results of method step1010-1070, which generates additional data to be stored on the database100, to determine the Weibull shape and scale parameters based on thecoupon-level testing data returned in steps 1020-1070. A further methodstep represented by a code segment 3000 applies the Weibull shapeparameters to generate an S-N curve for use in computing the LoadEnhancement Factor (LEF) for component level testing in step/codesegment 3000.

Since these equations require the fatigue life of the component to bedefined in terms of the number of lifetimes, the definition of 1lifetime must be equated to a corresponding cycle count, X₁. If thetraditional LEF approach is used, note that the static strength valueswill not be considered in the statistical analysis, whereas the methodpresented in Section 2000 will utilize the static data.

The stress level on the Mean S-N curve that corresponds to 1 lifetime,(S₁), is the applied stress that can be used in the component fatiguetest to achieve the desired levels of reliability and confidence.Equivalently, the component test may run at a stress level of S₂ for aduration equal to the Life Factor multiplied by 1 lifetime. Theresulting duration is equal to the Life Factor. Mathematically, the LifeFactor is defined by EQ-15 and computed in step 3020,

$\begin{matrix}{{N_{F} = \frac{\Gamma\left( \frac{\beta_{L} + 1}{\beta_{L}} \right)}{\left\lbrack \frac{- {\ln(R)}}{\frac{\chi_{\gamma,{2\; n_{0}}}^{2}}{2\; n_{0}}} \right\rbrack^{\frac{1}{\beta_{L}}}}}\;} & \left( {{EQ}\text{-}15} \right)\end{matrix}$Refer to method step 1050 in FIG. 2 for the values of the Chi-Squareddistribution. Note that the sample size n₀ being used now corresponds tothe number of fatigue tests planned at the component level, not thecoupon level.

If a large number of replicates are tested on the coupon-level, it isreasonable to assume that the Weibull shape parameters accuratelyquantify the fatigue scatter and these values are inherent properties ofthe material. To bridge the gap between coupon- and component-leveltesting, the sample size in the equations of this code segment mustaccount for the number of component tests since the component samplesize also affects the level of reliability. Therefore, the shapeparameters are developed using the number of coupon replicates, whilethe Life, Load, and Load Enhancement Factors computed in the ModifiedJoint Weibull Analysis component testing code segment 3000 rely on thenumber of components tested.

The Load Factor, (S_(F)), is then computed by EQ-16 in method step 3030and is represented as,

$\begin{matrix}{{S_{F} = \frac{\left\lbrack {K\left( N_{0} \right)} \right\rbrack{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}}{\left\lbrack \frac{- {\ln(R)}}{\frac{\chi_{\gamma,{2\; n_{0}}}^{2}}{2\; n_{0}}} \right\rbrack^{\frac{1}{\beta_{r}}}}}\;} & \left( {{EQ}\text{-}16} \right)\end{matrix}$Where the scaling coefficient, [K(N₀)], is defined as,

$\begin{matrix}{\left\lbrack {K\left( N_{0} \right)} \right\rbrack = \frac{\left\lbrack {\Gamma\left( \frac{\beta_{L} + 1}{\beta_{L}} \right)} \right\rbrack^{\frac{\beta_{L}}{\beta_{r}}}}{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}} & \left( {{EQ}\text{-}17} \right)\end{matrix}$This scaling coefficient is required to ensure that the test durationequals N_(F) when the Load Factor, S_(F)=1. Again, the Weibull shape andscale parameters are those previously derived in the Modified JointWeibull analysis code segment 1000. Note that the Load Factor, (S_(F)),can only be employed when the planned test duration of thecomponent-level fatigue test is assumed to equal 1 lifetime.

If both the applied loads and duration of the component-level fatiguetest are varied, then the Load Enhancement Factor (LEF) may be employed.This factor is computed as equation EQ-18 in method step 2070,

$\begin{matrix}{{LEF} = {\frac{\left\lbrack {K\left( N_{0} \right)} \right\rbrack{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}}{\left\lbrack \frac{{- {\ln(R)}}\left( N_{0} \right)^{\beta_{L}}}{\frac{\chi_{\gamma,{2\; n_{0}}}^{2}}{2\; n_{0}}} \right\rbrack^{\frac{1}{\beta_{r}}}} = \left( \frac{N_{F}}{N_{0}} \right)^{(\frac{\beta_{L}}{\beta_{r}})}}} & \left( {{EQ}\text{-}18} \right)\end{matrix}$Note that if the test duration is equal to 1 lifetime, N₀=1, the LoadEnhancement Factor is equal to the Load Factor, S_(F). Similarly, if theLoad Enhancement Factor is equal to 1, solving for the test durationwill yield the Life Factor.

The LEF is applied in a similar manner as the Load Factor, with theexception that the test duration must be specified. Whereas the LoadFactor is associated with a duration equal to 1 lifetime, a LoadEnhancement Factor uses a test duration that is not necessarily equal to1 lifetime. In lieu of using a duration of 1 lifetime and increasing theloads by a finite percentage, an infinite number of equivalentcombinations are possible by using the Load Enhancement Factor.

If the Modified Joint Weibull analysis will be used to analyze fatiguedata, it is necessary that at least two coupon replicates must be testedper stress level. In addition, at least two coupon replicates from theentire test program must survive to the run-out condition and be testedfor residual strength. These items are checked in the flow chart of FIG.1 in steps 200, 320, 340, and 420. As with the traditional approach tocompute Load Enhancement Factors, this modified approach does notaccount for variability in static strength. The use of scatter in staticstrength is incorporated in the Weibull Regression analysis method andcode segment 2000 presented below.

FIG. 4 is a flow chart of the Weibull Regression analysis segment of amethod 2000 according to an embodiment. Given the coupon test data withsingle stress levels, a Weibull Regression model can be used to estimatethe S-N relationship and provide estimates of various quantitiesassociated with the Load Enhancement Factor, as noted in the Whitehead,Vol. II reference and the traditional method for calculating LEF usingWeibull Regression modeling. As noted, the Weibull Regression analysisimproves on the traditional method and includes accommodation for bothsingle and multiple stress level coupon testing and the method utilizesthe static data.

Assuming that there are different fatigue stress levels and a Weibulldistribution is appropriate to describe the fatigue failure at eachstress level, this Weibull model with different scale parameters canhave the following form in equation EQ-19, where α_(i) is the scaleparameter for the i^(th) group, and β_(SN) is the common shapeparameter,

$\begin{matrix}{{f\left( x_{ij} \right)} = {\frac{\beta_{SN}}{\alpha_{i}}\left( \frac{x_{ij}}{\alpha_{i}} \right)^{\beta_{SN} - 1}{\exp\left( {- \left( \frac{x_{ij}}{\alpha_{i}} \right)^{\beta_{SN}}} \right)}}} & \left( {{EQ}\text{-}19} \right)\end{matrix}$where x_(ij) represents the failure cycles for the j^(th) coupon in thei^(th) group. It is conceivable that a_(i) changes with the fatiguestress level as a covariate.

At step 2010, the Weibull distribution and related power laws isapplied. The Weibull regression model representing the power law in theS-N curve would relate the log of the scale parameter with the log ofstress level in method step 2020. The underlying relationship may be oneof any number of linear or non-linear relationships between the log ofthe scale parameter and the log of stress level. A non-limiting exampleis found in a linear relationship between log of the scale parameter andlog of stress level, is expressed by equation EQ-20,

$\begin{matrix}{{\ln\left( \alpha_{i} \right)} = {\theta_{0} + {\theta_{1}{\ln\left( S_{i} \right)}}}} & \left( {{EQ}\text{-}20} \right)\end{matrix}$The method makes use of the relationship between Weibull and ExtremeValue distribution to implement an estimation procedure in method step2030 Let y_(ij)=ln(x_(ij)).

The probability density function and the cumulative distributions for yare represented in equations EQ-21 and EQ-22, respectively,

$\begin{matrix}{{g\left( y_{ij} \right)} = {\frac{1}{\sigma}{\exp\left( {\frac{y_{ij} - \phi_{i}}{\sigma} - {\exp\left( \frac{y_{ij} - \phi_{i}}{\sigma} \right)}} \right)}}} & \left( {{EQ}\text{-}21} \right)\end{matrix}$

$\begin{matrix}{{G\left( y_{ij} \right)} = {1 - {\exp\left( {- {\exp\left( \frac{y_{ij} - \phi_{i}}{\sigma} \right)}} \right)}}} & \left( {{EQ}\text{-}22} \right)\end{matrix}$where φ_(i)=θ₀+θ₁ ln(S_(i))=ln(α_(i)) and σ=1/β_(SN). Then, the loglikelihood function for Extreme Value distribution with covariate in Mgroups is defined in equations EQ-23 and EQ-24,

$\begin{matrix}{{\ln\; L} = {{\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}\left( {\frac{y_{ij} - \phi_{i}}{\sigma} - {\ln(\sigma)}} \right)}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\exp\left( \frac{y_{ij} - \phi_{i}}{\sigma} \right)}}}}} & \left( {{EQ}\text{-}23} \right)\end{matrix}$

$\begin{matrix}{{\ln\; L} = {{\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}\left( {\frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} - {\ln(\sigma)}} \right)}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\exp\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}}}}} & \left( {{EQ}\text{-}24} \right)\end{matrix}$where δ_(ij) is an indicator variable with two values. δ_(ij)=1 wheny_(ij) is a failure, and δ_(ij)=0 when y_(ij) is a run-out.

Setting the first partial derivatives of the above expression to zero,one obtains,

$\begin{matrix}{0 = {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\frac{- 1}{\sigma}\left( {\delta_{ij} - {\exp\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}} \right)}}}} & \left( {{EQ}\text{-}25} \right)\end{matrix}$

$\begin{matrix}{0 = {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\frac{- 1}{\sigma}\left( {{\delta_{ij}{\ln\left( S_{i} \right)}} - {{\ln\left( S_{i} \right)}{\exp\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}}} \right)}}}} & \left( {{EQ}\text{-}26} \right)\end{matrix}$

$\begin{matrix}{0 = {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{i}}{\frac{- 1}{\sigma}\left( {{\delta_{ij}\left\lbrack {\frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} + 1} \right\rbrack} - {{\exp\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}} \right)}}}} & \left( {{EQ}\text{-}27} \right)\end{matrix}$There is no helpful simplification for equations EQ-25 and EQ-26 interms of θ₀ and θ₁. Hence, solving for θ₀, θ₁, and σ requires anumerical algorithm, for example, but not limited to, the Newton-Rhapsonmethod. These parameters are estimated through nonlinear optimization inmethod step 2040.

Once these parameters are estimated through a nonlinear optimization,the 95% lower confidence limit for the 100q^(th) percentile, Y_(q)(S),is given by equation EQ-28 employed in method step 2050,

$\begin{matrix}{{{\hat{Y}}_{q}(S)} - {1.645\sqrt{V\left( {{\hat{Y}}_{q}(S)} \right)}}} & \left( {{EQ}\text{-}28} \right)\end{matrix}$where the 100 q^(th) percentile estimate at S is

Ŷ_(q)(S) = θ̂₀ + θ̂₁ln (S) + C_(q)σ̂,and its variance is approximated by EQ-29 in method step 2059,

$\begin{matrix}{{V\left( {{\hat{Y}}_{q}(S)} \right)} \approx {{V\left( {\hat{\theta}}_{0} \right)} + {{\ln(S)}^{2}{V\left( {\hat{\theta}}_{1} \right)}} + {C_{q}^{2}{V\left( \hat{\sigma} \right)}} + {2\;\ln\;(S){{Cov}\left( {{\hat{\theta}}_{0},{\hat{\theta}}_{1}} \right)}} + {2C_{q}{{Cov}\left( {{\hat{\theta}}_{0},\hat{\sigma}} \right)}} + {2{\ln(S)}C_{q}{{Cov}\left( {{\hat{\theta}}_{1},\hat{\sigma}} \right)}}}} & \left( {{EQ}\text{-}29} \right)\end{matrix}$where C_(q)=ln(−ln(1−q)). The value 1.645 is the 95^(th) percentile of astandard normal. This value will change if the confidence level change.The B-basis value is the 95% lower confidence limit in equation EQ-28with q=0.10, and A-basis with q=0.01. Note that the variance andcovariance estimates for these parameters are needed to complete thiscomputation.

The variance and covariance estimates for these parameters are computedin steps 2050-2060 from the elements of the inverse of the observedinformation matrix (I₀ ⁻¹). In this Extreme Value regression model, theobserved information matrix has the form provided in EQ-30,

$\begin{matrix}{I_{0} = \begin{bmatrix}{- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{0}}{\partial\theta_{0}}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{0}}{\partial\theta_{1}}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{0}}{\partial\sigma}}} \\{- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{0}}{\partial\theta_{1}}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{1}}{\partial\theta_{1}}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{1}}{\partial\sigma}}} \\{- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{0}}{\partial\sigma}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\theta_{1}}{\partial\sigma}}} & {- \frac{\partial^{2}{\ln(L)}}{{\partial\sigma}{\partial\sigma}}}\end{bmatrix}} & \left( {{EQ}\text{-}30} \right)\end{matrix}$Because the first partial derivatives are zero at the Maximum LikelihoodEstimation (MLE), a certain amount of simplification is possible for theinformation matrix when using this estimation technique. After somealgebraic manipulation, it has the following form.

$\begin{matrix}{I_{0} = {\frac{1}{{\hat{\sigma}}^{2}}\begin{bmatrix}{\sum\limits_{i = 1}^{m}\; n_{f_{i}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}{\ln\left( S_{i} \right)}}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}\left( {{\hat{z}}_{ij} + 1} \right)}}} \\{\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}{\ln\left( S_{i} \right)}}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{{\exp\left( {\hat{z}}_{ij} \right)}{\ln\left( S_{i} \right)}^{2}}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{{\exp\left( {\hat{z}}_{ij} \right)}{\hat{z}}_{ij}{\ln\left( S_{i} \right)}}}} \\{\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{\delta_{ij}\left( {{\hat{z}}_{ij} + 1} \right)}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}{{\exp\left( {\hat{z}}_{ij} \right)}{\hat{z}}_{ij}{\ln\left( S_{i} \right)}}}} & {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n_{i}}\left( {\delta_{ij} + {{\exp\left( {\hat{z}}_{ij} \right)}{\hat{z}}_{ij}^{2}}} \right)}}\end{bmatrix}}} & \left( {{EQ}\text{-}31} \right)\end{matrix}$where

ẑ_(ij) = (y_(ij) − θ̂₀ − θ̂₁ln (S_(i)))/σ̂.is the inverse of the observed information matrix

$\hat{\sum}\;{= I_{0}^{- 1}}$provided as equation EQ-32 or,

$\begin{matrix}{\hat{\Sigma} = {I_{0}^{- 1} = \begin{bmatrix}{V\left( {\hat{\theta}}_{0} \right)} & {{Cov}\left( {{\hat{\theta}}_{0},{\hat{\theta}}_{1}} \right)} & {{Cov}\left( {{\hat{\theta}}_{0},\hat{\sigma}} \right)} \\{{Cov}\left( {{\hat{\theta}}_{0},{\hat{\theta}}_{1}} \right)} & {V\left( {\hat{\theta}}_{1} \right)} & {{Cov}\left( {{\hat{\theta}}_{1},\hat{\sigma}} \right)} \\{{Cov}\left( {{\hat{\theta}}_{0},\hat{\sigma}} \right)} & {{Cov}\left( {{\hat{\theta}}_{1},\hat{\sigma}} \right)} & {V\left( \hat{\sigma} \right)}\end{bmatrix}}} & \left( {{EQ}\text{-}32} \right)\end{matrix}$Note that the B-basis value is a complex function of the stress levelthrough this variance function. It does not have the simple form as the“known” shape parameter under “known” shape parameter scenario. Thisdistinguishes the method of the Weibull Regression model from previouslydeveloped methodologies.

However, similar to the method used to develop of Load EnhancementFactor (LEF) under the traditional approach defined in Whitehead et al.,LEF is still defined as the ratio of two load values (S_(E), S_(B))times a scaling function. LEF is evaluated at N₀ lifetime is definedthrough equation EQ-33,

$\begin{matrix}{{{LEF}\left( N_{0} \right)} = {\frac{S_{E}}{S_{B}}{K\left( N_{0} \right)}}} & \left( {{EQ}\text{-}33} \right)\end{matrix}$Under the Weibull regression model described by equations EQ-19 andEQ-20, noted above, with a linear relationship in log-fatigue stresslevel, the expected value for fatigue failure cycle at a given stresslevel (S) is estimated at method step 2030 equation EQ-34 expressed as,

$\begin{matrix}{{E\left( X \middle| S \right)} = {{\exp\left( {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln(S)}}} \right)}{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)}}} & \left( {{EQ}\text{-}34} \right)\end{matrix}$and the B-basis value for fatigue failure cycle at a given stress level,S, is given by equation EQ-35 as,

$\begin{matrix}{{B\left( X \middle| S \right)} = {\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln(S)}} + {C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {\hat{Y}}_{0.10} \middle| S \right)}}} \right\rbrack}} & \left( {{EQ}\text{-}35} \right)\end{matrix}$If “one lifetime” is a pre-determined number of cycles derived from pastexperience on structures and usage, then let X₁ be the number of cyclesthat defines “one lifetime”. This can be provided by the user in methodstep 2050. Then, the method solves for S_(E) through the E(X|S), and forS_(B) through the B(X|S), using equations EQ-36 and EQ-37, as

$\begin{matrix}{X_{I} = {{E\left( {X❘S} \right)} = {{\exp\left( {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln(S)}}} \right)}{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)}}}} & \left( {{EQ} - 36} \right)\end{matrix}$which leads to an explicit form for S_(E),

$\begin{matrix}{S_{E} = {\exp\left( \frac{{\ln\left( X_{I} \right)} - {\ln\left( {\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)} - {\hat{\theta}}_{0}}{{\hat{\theta}}_{1}} \right)}} & \left( {{EQ}\text{-}37} \right)\end{matrix}$To find S_(B), one would solve for it in equation EQ-38, as

$\begin{matrix}{X_{I} = {\exp\left( {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{B} \right)}} + {C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}❘S_{B}} \right)}}} \right)}} & \left( {{EQ} - 38} \right)\end{matrix}$Note that the variance estimate involves stress setting. Hence, S_(B) isfound by a nonlinear solver in method step 2060.

Again, following the traditional procedure as outlined in Whitehead Vol.I, the Life Factor is necessary to construct LEF. Life factor is theratio of the expected failure cycle at S_(B) and the number of cyclesdefining one lifetime. The Life Factor described through equation EQ-39,

$\begin{matrix}{N_{F} = {\frac{E\left( X \middle| S_{B} \right)}{X_{I}} = \frac{{\exp\left( {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{B} \right)}}} \right)}{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)}}{X_{I}}}} & \left( {{EQ}\text{-}39} \right)\end{matrix}$As indicated above, S_(B) must satisfy the following equation EQ-40,

$\begin{matrix}{X_{I} = {{\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{B} \right)}}} \right\rbrack}{\exp\;\left\lbrack {{C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}❘S_{B}} \right)}}} \right\rbrack}}} & \left( {{EQ} - 40} \right)\end{matrix}$Therefore, the Life Factor under this Weibull regression approach can bewritten as equation EQ-41, which is computed in method step 2070,

$\begin{matrix}{N_{F} = \frac{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)}{\exp\left\lbrack {{C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}\left. S_{B} \right)} \right.}}} \right\rbrack}} & \left( {{EQ}\text{-}41} \right)\end{matrix}$If one chooses to work with A-basis instead of B-basis, one would findS_(A) by solving equation EQ-420 in place of equation EQ-40, as inmethod step 2075,

$\begin{matrix}{X_{I} = {\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{A} \right)}} + {C_{0.01}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.01}❘S_{A}} \right)}}} \right\rbrack}} & \left( {{EQ} - 42} \right)\end{matrix}$In order to determine the scaling function K( ) in equation EQ-33, thesame condition is imposed that was imposed from the traditional WeibullRegression analysis approach on LEF, namely, LEF(N_(F))=1. From equationEQ-33, applying this condition yields equation EQ-43,

$\begin{matrix}{{K\left( N_{F} \right)} = {\frac{S_{B}}{S_{E}} = \frac{S_{B}}{\exp\left\lbrack {\left\{ {{\ln\left( X_{I} \right)} - {\ln\left( {\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)} - {\hat{\theta}}_{0}} \right\}/{\hat{\theta}}_{1}} \right\rbrack}}} & \left( {{EQ}\text{-}43} \right)\end{matrix}$Substituting the definition of N_(F) in equation EQ-40 into K(N_(F)) andafter some algebraic manipulations, one would obtain equation EQ-44, as

$\begin{matrix}{{K\left( N_{F} \right)} = \left( N_{F} \right)^{1/{\hat{\theta}}_{1}}} & \left( {{EQ} - 44} \right)\end{matrix}$Then, replacing N_(F) with N₀ because they both are relative to X₁, thefinal form of LEF based on Weibull regression has the form of equationEQ-45 in method step 2075,

$\begin{matrix}{{{LEF}\left( N_{0} \right)} = {{\frac{S_{E}}{S_{B}}\left( N_{0} \right)^{1/{\hat{\theta}}_{1}}} = {\frac{\exp\left( {{- {\hat{\theta}}_{0}}/{\hat{\theta}}_{1}} \right)}{S_{B}}\left( \frac{X_{1}N_{0}}{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)^{1/{\hat{\theta}}_{1}}}}} & \left( {{EQ}\text{-}45} \right)\end{matrix}$Despite the fact that there is no analytical form for LEF here, it isfeasible to compute LEF using equations EQ-40 and EQ-45.

This Weibull Regression model is well documented in the statisticalliterature. Its application on LEF follows the same general philosophyas in the concept in Whitehead, et al. reference. Whitehead et al. didnot use any regression approach in computing LEF, namely they did notmake use of the S-N curve at all. Providing that LEF is a scaled ratioof a parameter (expected failure stress) and a parameter estimate (95%lower confidence limit for 10^(th) percentile=B-basis). However, theWeibull Regression model used to compute LEF incorporates the S-Nrelationship through the intercept and slope estimates ({circumflex over(θ)}₀, {circumflex over (θ)}₁) in the computation. The WeibullRegression model also employs sampling variation through the V(Ŷ_(q)(S))in finding S_(B), no such variation is accounted for in the traditionalLEF method. The definition of one lifetime in fatigue cycles is alsoexplicitly involved. The definition is implicitly used in thetraditional LEF methodology. This formulation of LEF in the WeibullRegression method does not involve residual strength. The WeibullRegression method also allows “run-out” and requires multiple stresslevels. In addition, the Weibull Regression method does not requiremultiple replicates per stress level, like its predecessor.

In requiring multiple stress levels, there is at least one exception tothis limitation. The Regression Analysis can still be used withoutmultiple stress levels if sufficient static strength data is available,as noted in steps 430 and 450 of FIG. 1. When static test results areavailable, the method incorporates that into fitting the S-Nrelationship, essentially drawing the S-N “line”. The contribution ofstatic test in the Weibull model relies on several assumptions. Theseinclude, but are not limited to, the static strength is the stresslevel, the cycles to failure is equal to 1, the random scatter of staticstrength can be ignored, and the failure mode is the same betweenfatigue cycle failure and static failure. In essence, these static testvalues would become part of the failure cycle data in estimating theWeibull regression model.

FIG. 6 is a block diagram of a computer network 6000 in which anembodiment may be implemented. As shown in FIG. 6, the computer network6000 includes, for example, a server 6010, workstations 6020, and 6030,a material testing device 6040, a database 100, and a network 6070. Thenetwork 8070 is configured to provide a communication path for eachdevice of the network 6070 to communicate with the other devices.Additionally, the computer network 6000 may be the Internet, a publicswitched telephone network, a local area network, private wide areanetwork, wireless network, and the like.

In various embodiments, a method of computing load enhancement factorsas described herein may be executed on the server 6010 and/or either orboth of the workstations 6020 and 6030. For example, in an embodiment ofthe disclosure, the server 6010 is configured to execute the method ofcomputing load enhancement factors, provide output for display to theworkstations 6020 and/or 6030, and receive input from the database 100,material testing device 6040, workstations 6020 and/or 6030. In variousother embodiments, one or both of the workstations 6020 and 6030 may beconfigured to execute the method of computing load enhancement factorsindividually or co-operatively.

The material testing device 6040 may be configured to preform couponand/or component testing and output any suitable results in a computerreadable format. Additionally, data associated with coupon testing,component testing, and the like, may be stored on the database 100. Thedatabase 100 may additionally be configured to receive and/or forwardsome or all of the stored data. Moreover, in yet another embodiment,some or all of the computer network 6000 may be subsumed within a singledevice.

Although FIG. 6 depicts a computer network, it is to be understood thatthe various embodiments are not limited to operation within a computernetwork, but rather, the some or all of the embodiments may be practicedin any suitable electronic device. Accordingly, the computer networkdepicted in FIG. 6 is for illustrative purposes only and thus is notmeant to limit the various embodiments in any respect.

The many features and advantages of the various embodiments are apparentfrom the detailed specification, and thus, it is intended by theappended claims to cover all such features and advantages that fallwithin the true spirit and scope of these and other embodiments.Further, since numerous modifications and variations will readily occurto those skilled in the art, it is not desired to limit the embodimentsto the exact construction and operation illustrated and described, andaccordingly, all suitable modifications and equivalents may be resortedto, falling within the scope of the various embodiments.

1. A method of computing at least one of a Load Factor, a Life Factorand a Load Enhancement Factor using Modified Joint Weibull Analysis,comprising the steps of: retrieving a test data set from at least onedatabase; analyzing the retrieved test data of the test data set for fitwith a Weibull distribution model for the test data; analyzing theretrieved test data to determine if at least two coupons have beentested and if applied loads and duration of testing at a component-levelwere varied; computing at least one shape parameter for the Weibulldistribution model and storing the at least one shape parameter;computing at least one scale parameter for the Weibull distributionmodel of the test data and storing the at least one scale parameter;computing a stress to life cycle relationship accounting for scatter inthe test data through the Weibull distribution model of the test data byrelating a log of the scale parameter of the Weibull distribution modelwith a log of a stress level of a stress to life cycle relationship; andcomputing, using a processor, the at least one of a Life Factor, a LoadFactor and a Load Enhancement Factor as a function of a slope and anintercept of the computed stress to life cycle relationship, therebyaccounting for scatter in the at least one Life Factor, a Load Factorand a Load Enhancement Factor.
 2. The method of claim 1, wherein thestep of computing at least one shape parameter further comprisescomputing the Weibull shape parameter value for fatigue life, (β_(L)),using the equation${{\sum\limits_{i = 1}^{M}\;\left( {n_{f_{i}}\frac{\sum\limits_{j = 1}^{n_{i}}\;{X_{ij}^{\beta_{L}}{\ln\left( X_{ij} \right)}}}{\sum\limits_{j = 1}^{n_{i}}\; X_{ij}^{\beta_{L}}}} \right)} - {\sum\limits_{i = 1}^{M}\frac{n_{f_{i}}}{\beta_{L}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{fi}}{\ln\left( X_{ij} \right)}}}} = 0$wherein: i represents a Group number corresponding to a stress level forcoupon-level testing; M represents a total number of groups forcoupon-level testing; n_(i) represents a number of coupons in an i^(th)group; n_(f) _(i) represents a number of coupons failing in fatigue inthe i^(th) group; j represents a data point element within i^(th) group;X represents a number of fatigue cycles to failure for a coupon; andβ_(L) represents a shape parameter of a Weibull model for fatigue lifemeasured from coupon-level testing.
 3. The method of claim 2, whereinthe step of computing at least one shape parameter further comprisescomputing the Weibull shape parameter value for residual strength,(β_(r)), using the equation${{\sum\limits_{i = 1}^{M}\;\left( {n_{r_{i}}\frac{\sum\limits_{j = 1}^{n_{i}}\;{W_{ij}^{\beta_{r}}{\ln\left( W_{ij} \right)}}}{\sum\limits_{j = 1}^{n_{i}}\; W_{ij}^{\beta_{r}}}} \right)} - {\sum\limits_{i = 1}^{M}\frac{n_{r_{i}}}{\beta_{r}}} - {\sum\limits_{i = 1}^{M}{\sum\limits_{j = 1}^{n_{r_{i}}}{\ln\left( W_{ij} \right)}}}} = 0$wherein: W represents a residual strength at X run-out cycle and stressS; n_(r) _(i) represents the number of coupons used in the Weibull modelfor residual strength in the i^(th) group; and β_(r) represents a shapeparameter of the Weibull model for residual strength measured fromcoupon-level testing.
 4. The method of claim 3, wherein the step ofcomputing at least one shape parameter further comprises computing theWeibull shape parameter value for static strength, (β_(s)), is computedin equation${{n_{s}\frac{\sum\limits_{j = 1}^{n_{s}}\;{S_{j}^{\beta_{s}}{\ln\left( S_{j} \right)}}}{\sum\limits_{j = 1}^{n_{s}}\; S_{j}^{\beta_{s}}}} - \frac{n_{s}}{\beta_{s}} - {\sum\limits_{j = 1}^{n_{s}}{\ln\left( S_{j} \right)}}} = 0$wherein: β_(s) represents a shape parameter of the Weibull model forstatic strength measured from coupon-level testing; n_(s) represents anumber of coupons tested for static strength; and S_(j) represents astatic strength of j data point for coupon-level testing.
 5. The methodof claim 4, wherein the step of computing at least one shape parameterfurther comprises solving for the Weibull shape parameter for staticstrength, Weibull shape parameter for residual strength, and Weibullshape parameter for fatigue life values for equations using an iterativeapproach.
 6. The method of claim 5, wherein the step of computing atleast one scale parameter further comprises calculating the Weibull lifescale parameter for each stress level using equation$\alpha_{L_{i}} = \left\lbrack {\frac{1}{n_{f_{i}}}{\sum\limits_{j = 1}^{n_{i}}X_{ij}^{\beta_{L}}}} \right\rbrack^{\frac{1}{\beta_{L}}}$wherein: α_(L) _(i) represents a scale parameter of the Weibull modelfor fatigue life measured from coupon-level testing for i^(th) group. 7.The method of claim 6, wherein the step of computing at least one scaleparameter further comprises calculating the Weibull scale parameter forresidual strength using equation$\alpha_{r_{i}} = \left\lbrack {\frac{1}{n_{r_{i}}}{\sum\limits_{j = 1}^{n_{i}}W_{ij}^{\beta_{r}}}} \right\rbrack^{\frac{1}{\beta_{r}}}$wherein: α_(r) _(i) represents the scale parameter of the Weibull modelfor residual strength measured from coupon-level testing for the i^(th)group (stress level).
 8. The method of claim 7, wherein the step ofcomputing at least one scale parameter further comprises calculatingscale parameter for static strength using equation$\alpha_{s} = \left\lbrack {\frac{1}{n_{s}}{\sum\limits_{j = 1}^{n_{s}}S_{j}^{\beta_{s}}}} \right\rbrack^{\frac{1}{\beta_{s}}}$wherein: α_(s) represents the scale parameter of the Weibull model forstatic strength measured from coupon-level testing.
 9. The method ofclaim 1, wherein the step of computing a stress to life cyclerelationship accounting for scatter further comprises obtaining valuesfor Chi-squared distribution and Gamma functions.
 10. The method ofclaim 9, wherein the step of computing a stress to life cyclerelationship accounting for scatter further comprises the step ofobtaining user input for a desired confidence level (γ), calculating thevalue of the Chi-squared distribution for the obtained level ofconfidence (χ_(γ,2n) _(L) ²) wherein 2n_(L) are the degrees of freedomcorresponding to coupon-level test data.
 11. The method of claim 10,wherein the step of obtaining values for the Chi-squared distributionand Gamma functions further comprises calculating values of the Gammafunction for the life shape parameter using the equation:$\Gamma\left\lbrack \left( \frac{\beta_{L} + 1}{\beta_{L}} \right) \right\rbrack$wherein: Γ represents a gamma function; and β_(L) represents the shapeparameter of the Weibull model for fatigue life measured fromcoupon-level testing.
 12. The method of claim 11, wherein the step ofobtaining values for the Chi-squared distribution and Gamma functionsfurther comprises calculating the values of the Gamma function for theresidual strength shape parameters using the equation:$\Gamma\left\lbrack \left( \frac{\beta_{r} + 1}{\beta_{r}} \right) \right\rbrack$wherein: β_(r) represents the shape parameter of the Weibull model forresidual strength measured from coupon-level testing.
 13. The method ofclaim 12, wherein the step of obtaining values for the Chi-squareddistribution and Gamma functions further comprises storing values forthe Gamma function for varying shape parameters in a database of Gammafunction values.
 14. The method of claim 13, wherein the step ofobtaining values for the Chi-squared distribution and Gamma functionsfurther comprises a lookup step, wherein the value of the chi square andgamma functions are determined by the lookup step in a table of storedvalues in the database.
 15. The method of claim 14, wherein the step ofcomputing a stress to life cycle relationship accounting for scatterfurther comprises, calculating mean static strength using the equation:$S_{M} = {\alpha_{s}{\Gamma\left\lbrack \left( \frac{\beta_{s} + 1}{\beta_{s}} \right) \right\rbrack}}$wherein: S_(M) represents mean static strength; α_(s) represents a scaleparameter of a Weibull model for static strength measured fromcoupon-level testing; and β_(s) represents a shape parameter of theWeibull model for static strength measured from coupon-level testing.16. The method of claim 15, wherein the step of computing a stress tolife cycle relationship accounting for scatter further comprisescalculating mean residual strength using the equation:$P_{T_{i}} = {\alpha_{r_{i}}{\Gamma\left\lbrack \left( \frac{\beta_{r} + 1}{\beta_{r}} \right) \right\rbrack}}$wherein: P_(T) _(i) represents a mean residual strength per stresslevel; α_(r) _(i) represents the scale parameter of the Weibull modelfor residual strength measured from coupon-level testing for the i^(th)group.
 17. The method of claim 16, wherein the step of computing astress to life cycle relationship accounting for scatter furthercomprises calculating mean fatigue life using equation:$N_{M_{i}} = {\alpha_{L_{i}}{\Gamma\left\lbrack \left( \frac{\beta_{L} + 1}{\beta_{L}} \right) \right\rbrack}}$wherein: N_(M) _(i) =E(X/S) represents the mean life for each stresslevel for coupon-level testing; α_(Li) represents the scale parameter ofthe Weibull model for fatigue life measured from coupon-level testingfor the i^(th) group.
 18. The method of claim 17, wherein the step ofcomputing a stress to life cycle relationship accounting for scatterfurther comprises obtaining a reliability rating and calculating thedesign life at a desired reliability and the residual strength allowableat the reliability computed using the equations:$N_{i} = {{\alpha_{L_{i}}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2n_{L}}}^{2}}{2n_{L}}} \right\rbrack}^{\frac{1}{\beta_{L}}}\mspace{14mu}{and}}$$W_{r_{i}} = {\alpha_{r_{i}}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2n_{r}}}^{2}}{2n_{r}}} \right\rbrack}^{\frac{1}{\beta_{r}}}$wherein: N_(i) represents a design life at a desired reliability for thei^(th) stress level; R represents a reliability function; χ_(γ,ν) ²represents a Chi-square random variable with γ confidence level and νdegrees of freedom; n_(L) represents a total number of coupons used in aWeibull model for fatigue life cycle; and n_(r) represents a totalnumber of coupons used in the Weibull model for residual strength. 19.The method of claim 18, wherein the step of computing a stress to lifecycle relationship accounting for scatter further comprises calculatingstatic strength allowable at the reliability using the equation:$S_{r} = {\alpha_{s}\left\lbrack \frac{\left\lbrack {- {\ln(R)}} \right\rbrack}{\frac{\chi_{\gamma,{2n_{s}}}^{2}}{2n_{s}}} \right\rbrack}^{\frac{1}{\beta_{s}}}$and the sample size in the denominator refers to the total number ofcoupons reaching the run-out condition; wherein: S_(r) represents thestatic strength allowable; n_(s) represents a number of coupons testedfor static strength; α_(s) represents the scale parameter of the Weibullmodel for static strength measured from coupon-level testing.
 20. Themethod of claim 19, wherein the step of computing at least one of a LifeFactor, a Load Factor and a Load Enhancement Factor further comprisesthe step of computing the Life Factor using the equation:$N_{F} = \;\frac{\Gamma\left( \frac{\beta_{L} + 1}{\beta_{L}} \right)}{\left\lbrack \frac{{- {\ln(R)}}\;}{\frac{\chi_{\gamma,{2n_{0}}}^{2}}{2n_{0}}} \right\rbrack^{\frac{1}{\beta_{s}}}}$wherein the stored values of the Chi-Squared distribution are retrievedfrom a database of Gamma function values and the sample size being usedcorresponds to the number of fatigue tests planned on a component level;wherein: N_(F) represents the Life Factor; and n₀ represents a number ofcomponent-level tests.
 21. The method of claim 20, wherein the step ofcomputing at least one of a Life Factor, a Load Factor and a LoadEnhancement Factor further comprises the step of computing the LoadFactor, (S_(F)) using the equation:$S_{F} = \;\frac{\left\lbrack {K\left( N_{0} \right)} \right\rbrack{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}}{\left\lbrack \frac{{- {\ln(R)}}\;}{\frac{\chi_{\gamma,{2n_{0}}}^{2}}{2n_{0}}} \right\rbrack^{\frac{1}{\beta_{r}}}}$wherein the scaling coefficient, [K(N₀)], is computed in the form,$\left\lbrack {K\left( N_{0} \right)} \right\rbrack = \frac{\left\lbrack {\Gamma\left( \frac{\beta_{L} + 1}{\beta_{L}} \right)} \right\rbrack^{\frac{\beta_{L}}{\beta_{r}}}}{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}$wherein: S_(F) represents the load factor; N₀ represents a test durationin number of lifetimes.
 22. The method of claim 21, wherein the step ofcomputing at least one of a Life Factor, a Load Factor and a LoadEnhancement Factor further comprises the step of computing a LoadEnhancement Factor using the equation:${LEF} = {\frac{\left\lbrack {K\left( N_{0} \right)} \right\rbrack{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}}{\left\lbrack \frac{{{- {\ln(R)}}\left( N_{0} \right)^{\beta_{L}}}\;}{\frac{\chi_{\gamma,{2n_{0}}}^{2}}{2n_{0}}} \right\rbrack^{\frac{1}{\beta_{r}}}} = \left( \frac{N_{F}}{N_{0}} \right)^{(\frac{\beta_{L}}{\beta_{r}})}}$wherein the scaling coefficient, [K(N₀)], is computed in the form,$\left\lbrack {K\left( N_{0} \right)} \right\rbrack = \frac{\left\lbrack {\Gamma\left( \frac{\beta_{L} + 1}{\beta_{L}} \right)} \right\rbrack^{\frac{\beta_{L}}{\beta_{r}}}}{\Gamma\left( \frac{\beta_{r} + 1}{\beta_{r}} \right)}$wherein: LEF represents the Load Enhancement Factor.
 23. A method ofcalculating a Load Enhancement Factor for a composite structure usingWeibull Regression Analysis, comprising the steps of: obtaining coupontest data of a material from which the composite structure is formed;applying a Weibull distribution regression function to the coupon testdata to develop an estimated stress to fatigue life curve and applyingan equation to relate the log of a scale parameter of the Weibulldistribution regression function with the log of a stress level of thestress to fatigue life curve; implementing an estimation procedure toincorporate a stress to fatigue life relationship in the calculation ofthe Load Enhancement Factor through an intercept and a slope of theestimates of the stress to fatigue life curve; and specifying, by auser, a confidence level defined in a stress level of a fatigue cycledistribution; and calculating, using a processor, the Load EnhancementFactor using a variance and a co-variance matrix associated with theconfidence level and representing the intercept and the slope of thestress to fatigue life curve and developed from the estimation procedureand wherein the Load Enhancement Factor is a scaled ratio of an expectedfailure stress and the stress level upon which the confidence level isbased.
 24. The method of claim 23, wherein the coupon test data is froma single stress level.
 25. The method of claim 23, wherein the coupontest data is from multiple stress levels and includes static strengthdata.
 26. The method of claim 23, wherein the Weibull distributionregression function is defined by the equation:${{f\left( x_{ij} \right)} = {\frac{\beta_{SN}}{\alpha_{i}}\left( \frac{x_{ij}}{\alpha_{i}} \right)^{\beta_{SN} - 1}{\exp\left( {- \left( \frac{x_{ij}}{\alpha_{i}} \right)^{\beta_{SN}}} \right)}}},$wherein: α_(i) is the scale parameter for the i^(th) group; β_(SN) isthe common shape parameter; x_(ij) represents the failure cycles for thej^(th) coupon in the i^(th) group.
 27. The method of claim 26, whereinthe equation to relate the log of a scale parameter of the Weibulldistribution regression function with the log of a stress level of thestress to fatigue life curve is a linear relationship between a log ofthe scale parameter and a log of stress level that is establishedthrough the equation:ln(α_(i))=θ₀+θ₁ ln(S _(i)), wherein α_(i) is the scale parameter for thei^(th) group, θ₀ represents the intercept parameter of a Weibullregression model, θ₁ represents the slope parameter of the Weibullregression model, and S represents the stress level applied in fatiguetest or static strength for the i^(th) group.
 28. The method of claim27, wherein the step of implementing an estimation procedure furthercomprises implementing an estimation procedure with y_(ij)=ln(x_(ij)),and where the probability density function and the cumulativedistributions for y are computed by equations:${g\left( y_{ij} \right)} = {\frac{1}{\sigma}{\exp\left( {\frac{y_{ij} - \phi_{i}}{\sigma} - {\exp\left( \frac{y_{ij} - \phi_{i}}{\sigma} \right)}} \right)}}$${G\left( y_{ij} \right)} = {1 - {\exp\left( {- {\exp\left( \frac{y_{ij} - \phi_{i}}{\sigma} \right)}} \right)}}$with φ_(i)=θ₀+θ₁ ln(S_(i))=ln(α_(i)) and σ=1/β_(SN) and where the loglikelihood function for Extreme Value distribution is expressed with thefollowing equation, with covariate in M groups is described by theequation:${{\ln\; L} = {{\sum\limits_{i = 1}^{M}\;{\sum\limits_{j = 1}^{n_{i}}\;{\delta_{ij}\left( {\frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} - {\ln(\sigma)}} \right)}}} - {\sum\limits_{i = 1}^{M}\;{\sum\limits_{j = 1}^{n_{i}}\;{\exp\left( \frac{y_{ij} - \theta_{0} - {\theta_{1}{\ln\left( S_{i} \right)}}}{\sigma} \right)}}}}},$wherein δ_(ij) is an indicator variable with two values, δ_(ij)=1 wheny_(ij) is a failure, and δ_(ij)=0 when y_(ij) is a run-out and where theparameters θ₀ and θ₁ and σ are estimated through nonlinear optimization.29. The method of claim 28, wherein the step of implementing anestimation procedure further comprises computing a variance and acovariance estimate for θ₀, θ₁, and σ from the inverse of an observedinformation matrix, ${\hat{\sum}\;{= I_{0}^{- 1}}},$ provided by theequation: $\hat{\sum}{= {I_{0}^{- 1} = \begin{bmatrix}{V\left( \hat{\left. \theta_{0} \right)} \right.} & {{Cov}\left( \hat{\theta_{0},\hat{\left. \theta_{1} \right)}} \right.} & {{Cov}\left( \hat{\theta_{0},\hat{\left. \sigma \right)}} \right.} \\{{Cov}\left( \hat{\theta_{0},\hat{\left. \theta_{1} \right)}} \right.} & {V\left( \hat{\left. \theta_{1} \right)} \right.} & {{Cov}\left( \hat{\theta_{1},\hat{\left. \sigma \right)}} \right.} \\{{Cov}\left( \hat{\theta_{0},\hat{\left. \sigma \right)}} \right.} & {{Cov}\left( \hat{\theta_{1},\hat{\left. \sigma \right)}} \right.} & {V\left( \hat{\left. \sigma \right)} \right.}\end{bmatrix}}}$ wherein: I₀ ⁻¹ represents the observed informationmatrix; V represents a variance; Cov represents a co-variance; θ₀represents an intercept parameter of the Weibull distribution regressionfunction; θ₁ represents a slope parameter of the Weibull distributionregression function; σ=1/β_(SN).
 30. The method of claim 29, wherein thestep of obtaining a confidence level further comprises establishing anA-basis value of a 95% lower confidence limit for the 100q^(th)percentile, Y_(q)(S), using equation:${{{\hat{Y}}_{q}(S)} - {1.645\sqrt{V\left( {{\hat{Y}}_{q}(S)} \right)}}},$wherein the 100 q^(th) percentile estimate at S isŶ_(q)(S) = θ̂₀ + θ̂₁ln (S) + C_(q)σ̂, and a variance of S is approximatedby the expression:V(Ŷ_(q)(S)) ≈ V(θ̂₀) + ln (S)²V(θ̂₁) + C_(q)²V(σ̂) + 2ln (S)Cov(θ̂₀, θ̂₁) + 2C_(q)Cov(θ̂₀, σ̂) + 2ln (S)C_(q)Cov(θ̂₁, σ̂),wherein: C_(q)=ln(−ln(1−q)); and S represents a stress level applied ina fatigue test Y_(q)(S) represents a confidence limit at stress S forthe 100q^(th) percentile.
 31. The method of claim 29, wherein the stepof obtaining a confidence level further comprises establishing a B-basisvalue as a 95% lower confidence limit expressed as equation:${{\hat{Y}}_{q}(S)} - {1.645\sqrt{V\left( {{\hat{Y}}_{q}(S)} \right)}}$wherein: 1.645 represents a 95^(th) percentile of a standard normal. 32.The method of claim 29, further comprising the step of obtaining userinput for duration of one lifetime of the composite structure.
 33. Themethod of claim 29, wherein the step of computing the Load EnhancementFactor further comprises evaluating Load Enhancement Factor at N₀lifetime being defined through the equation:${{LEF}\left( N_{0} \right)} = {\frac{S_{E}}{S_{B}}{K\left( N_{0} \right)}}$with a linear relationship in log-fatigue stress level and the expectedvalue for fatigue failure cycle at a given stress level (S) beingestimated by the equation:${E\left( {X❘S} \right)} = {\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln(S)}} + {C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}❘S} \right)}}} \right\rbrack}$and the B-basis value for fatigue failure cycle at a given stress level,S, is computed by equation:${B\left( {X❘S} \right)} = {\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln(S)}} + {C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}❘S} \right)}}} \right\rbrack}$wherein: E(X/S) represents an expected fatigue failure cycle at stressS; B(X/S) represents a B-basis value in residual strength at X run-outcycles and stress S; X represents a number of fatigue cycles to failurefor a coupon; S_(E) represents a stress level computed from an expectedvalue of fatigue cycle distribution; S_(B) represents a stress levelcomputed from a B-basis value of fatigue cycle distribution; Krepresents a scaling function for the N₀ lifetime; F represents a gammafunction; V represents a variance; and (Y_(0.10)|S) represents aconfidence limit at stress S for the 100q ^(th) percentile with q=0.10.34. The method of claim 33, wherein the step of computing the LoadEnhancement Factor further comprises solving for S_(E) through theequation:$S_{E} = \;{\exp\left( \frac{{\ln\left( X_{I} \right)} - {\ln\left( {\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)} - {\hat{\theta}}_{0}}{{\hat{\theta}}_{1}} \right)}$wherein: X₁ being the number of cycles that defines one lifetimes θ₀represents the intercept parameter of the Weibull regression model; andθ₁ represents the slope parameter of the Weibull regression model. 35.The method of claim 34, wherein the step of computing the LoadEnhancement Factor further comprises solving for S_(B) through theequation:$X_{I} = {{\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{B} \right)}}} \right\rbrack}{\exp\left\lbrack {{C_{0.10}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.10}❘S_{B}} \right)}}} \right\rbrack}}$and using a nonlinear solver on the equation wherein: θ₀ represents anintercept parameter of the Weibull regression model; θ₁ represents aslope parameter of the Weibull regression model; S_(B) represents astress level computed from a B-basis value of fatigue cycledistribution; σ=1/β_(SN); (Y_(0.10)|S) represents a confidence limit atstress S for the 100q^(th) percentile with q=0.10; andC_(q)=ln(−ln(1−q)).
 36. The method of claim 35, wherein the computationof the Load Enhancement Factor includes calculating a Life factordescribed in equation:${N_{F} = \frac{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)}{\exp\left\lbrack {{C_{0.10}\hat{\sigma}} - {1.645\sqrt{\left. {{V\left( {\hat{Y}}_{0.10} \right.}S_{B}} \right)}}} \right\rbrack}}\;$and calculating S_(A) by solving equation:$X_{I} = {\exp\left\lbrack {{\hat{\theta}}_{0} + {{\hat{\theta}}_{1}{\ln\left( S_{A} \right)}} + {C_{0.01}\hat{\sigma}} - {1.645\sqrt{V\left( {{\hat{Y}}_{0.01}❘S_{A}} \right)}}} \right\rbrack}$and imposing LEF(N_(F))=1, the computation computing the ratio under thefollowing equations:${K\left( N_{F} \right)} = {\frac{S_{B}}{S_{E}} = \frac{S_{B}}{\exp\left\lbrack {\left\{ {{\ln\left( X_{I} \right)} - {\ln\left( {\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)} - {\hat{\theta}}_{0}} \right\}/{\hat{\theta}}_{1}} \right\rbrack}}$and K(N_(F))=(N_(F))¹/{circumflex over (θ)} ¹ and the computation of LEFbased on Weibull regression being expressed by the equation:${{{LEF}\left( N_{0} \right)} = {{\frac{S_{E}}{S_{B}}\left( N_{0} \right)^{1/{\hat{\theta}}_{1}}} = {\frac{\exp\left( {{- {\hat{\theta}}_{0}}/{\hat{\theta}}_{1}} \right)}{S_{B}}\left( \frac{X_{I}N_{0}}{\Gamma\left( {1 + {1/{\hat{\beta}}_{SN}}} \right)} \right)^{1/{\hat{\theta}}_{1}}}}},$wherein LEF is computed using the equation for LEE (N₀) and the equationfor S_(B) wherein: S_(A) represents a stress level computed from anA-basis value of a fatigue cycle distribution; K(N_(F)) represents ascaling function for Life Factor N_(F); and K(N₀) represents a scalingfor given lifetime N₀.
 37. A method of evaluating a composite componentof an aircraft, comprising the steps of: calculating a Load EnhancementFactor of the composite component of the aircraft using Modified JointWeibull Analysis, wherein calculating the Load Enhancement Factorcomprises the steps of: retrieving a test data set from at least onedatabase; analyzing the retrieved test data of the test data set for fitwith a Weibull regression model; computing at least one scale parameterfor the Weibull regression model; calculating a stress to life cyclerelationship accounting for scatter in the retrieved test data throughthe Weibull regression model of the retrieved test data by relating alog of the scale parameter of the Weibull regression model with a log ofa stress level of a stress to life cycle relationship; and calculating,using a processor, the Load Enhancement Factor as a function of a slopeand an intercept of the stress to life cycle relationship.